A new class of translation planes constructed by hyper-regulus replacement (Q1021310)

A class of hyper-reguli is determined in a Desarguesian affine plane that are not André hyper-reguli. It is considered the class of translation planes obtained by the replacement of such a hyper-regulus and shown that it is obtained a new class of translation planes, which are not André or generalized André. These planes have a very small affine homology group, whereas in the André or generalized André planes the affine homology group is rather larger. In fact, when \(n\) and \(q\) are both greater then 3, and \(n\) is odd, the affine homology group of these planes is trivial.